Corners and Records of the Poisson Process in Quadrant

TL;DR

This paper explores the relationship between the scale-invariant spacings lemma for Poisson processes and the properties of extreme points in a unit Poisson process within the positive quadrant, revealing new structural insights.

Contribution

It establishes a novel connection between the scale-invariant spacings lemma and the extremal structure of the Poisson process in the quadrant, enhancing understanding of Poisson process geometry.

Findings

Identifies a link between spacings and extreme points in Poisson processes.

Provides new insights into the structure of Poisson processes in the quadrant.

Connects existing lemmas with geometric properties of Poisson points.

Abstract

The scale-invariant spacings lemma due to Arratia, Barbour and Tavar{\'e} establishes the distributional identity of a self-similar Poisson process and the set of spacings between the points of this process. In this note we connect this result with properties of a certain set of extreme points of the unit Poisson process in the positive quadrant.